The isomorphism problem for classes of computable fields
Wesley Calvert
备注：
15 pages, updated references, general revision, previously entitled "The isomorphism problem for familiar classes of computable structures I", long version of published paper
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable models is very complex become classifiable when we consider only computable members.
We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non- classifiable. For some classes (undirected graphs, fields of fixed characteristic, and real closed fields) we show that the isomorphism problem is Σ^1_1 complete (the maximum possible), and for others it is of relatively low complexity. For instance, for algebraically closed fields, archimedean real closed fields, and vector spaces, we show that the isomorphism problem is Π^0_3 complete.
We show that every countable ideal of degrees that are low for isomorphism is contained in a principal ideal of degrees that are low for isomorphism by adapting an exact pair construction. We further show that within the hyperimmune-free degrees, lowness for isomorphism is entirely independent of computable traceability.
Computable Isomorphisms for Certain Classes of Infinite Graphs
我们研究了（2，1）：1结构，它由一个可数集$A$和一个函数$f: A \to A$组成，使得对于$A$中的每个元素$x$，$f$将$A$中的一个元素或两个元素映射到$x$ .这些结构扩展了Cenzer、Harizanov和Remmel研究的注入结构、2：1结构和（2，0）：1结构的概念，所有这些结构都可以被认为是无限有向图。我们研究了（2，1）：1结构的各种可计算性理论性质，最值得注意的是可计算范畴性。我们说一个结构$\mathcal{A}$是可计算范畴的，如果在$\mathcal{A}$的任何两个可计算副本之间存在可计算同构。给出了（2，1）：1结构可计算范畴的一个充分条件，并给出了具有不同可计算性理论性质的（2，1）：1结构的一些例子.
We contribute to the program of extending computable structure theory to the realm of metric structures by investigating lowness for isometric isomorphism of metric structures. We show that lowness for isomorphism coincides with lowness for isometric isomorphism and with lowness for isometry of metric spaces. We also examine certain restricted notions of lowness for isometric isomorphism with respect to fixed computable presentations, and, in this vein, we obtain classifications of the degrees that are low for isometric isomorphism with respect to the standard copies of certain Lebesgue spaces…
This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called "computable reducibility"), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe for instance that every ce…
A general theme of computable structure theory is to investigate when structures have copies of a given complexity $Γ$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a $Π^0_1$ equivalence structure with no $Σ^0_1$ copy, and in fact that the isomorphism types realized by the $Π^0_1$ equivalence structures coincide with those realized by the $Δ^0_2$ equivalence structures. We also construct a $Σ^0_1$ preorder with no $Π^0_1$ copy.
Recursively Enumerably Representable Classes and Computable Versions of the Fundamental Theorem of Statistical Learning
Institute for Symbolic Artificial Intelligence, Johannes Kepler University Linz（约翰内斯·开普勒大学林茨分校符号人工智能研究所）；Institute for the Interdisciplinary Study of Language Evolution, University of Zurich（苏黎世大学语言演化跨学科研究所）；Department of Mathematics and Statistics, University of Konstanz（康斯坦茨大学数学与统计学系）
In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable structures in two such classes, the classes of linear orders and Boolean algebras. We show that if $\mathcal L$ is a computable linear order of Hausdorff rank $n$, then for every bi-embeddable copy of it there is an embedding computable in $2n-1$ jumps from the atomic diagrams. We furthermore show that this is the best one can do: Let $\mathcal L$ be a computable linear…
In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems: whether a generated class is a real closed field and whether there exists a computable presentation of a generated class. We prove a series of theorems that lead to the result that there are no computable presentations neither for polynomial time computable no even for $E_n$-computable real numbers, where $E_n$ is a level in Grzegorczyk hierarchy, $n \geq 2$…
In this note we axiomatize the classes of rudimentary functions, primitive recursive functions, safe recursive set functions, and predicatively computable functions.
